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When is the energy of the 1D damped Klein-Gordon equation decaying?

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Abstract

We study time decay of the energy for the one dimensional damped Klein-Gordon equation. We give an explicit necessary and sufficient condition on the continuous damping function \(\gamma \ge 0\) for which the energy

$$\begin{aligned} E(t)=\int _{-\infty }^\infty |u_x|^2+|u|^2+ |u_t|^2 dx \end{aligned}$$

decays, whenever \((u(0), u_t(0))\in H^2(\mathbb R)\times H^1(\mathbb R)\).

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Notes

  1. See Proposition 1.

  2. Here, we depart from the usual definition, where eigenvalues of infinite multiplicities are considered as part of the essential spectrum. We will see though, that since eigenvalues do not appear in our setup, at least on the important set \(\sigma (\mathcal A)\cap i {\mathbb R}\), this is not consequential.

  3. The constants N and \(\kappa \) have subscript \(\gamma \), however we will remove this in the rest of the presentation.

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Authors and Affiliations

  1. Department of Mathematics, University of Kansas, 1460 Jayhawk Boulevard, Lawrence, KS, 66045–7523, USA

    Satbir Malhi & Milena Stanislavova

Authors
  1. Satbir Malhi
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  2. Milena Stanislavova
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Correspondence to Milena Stanislavova.

Additional information

Communicated by Loukas Grafakos.

Stanislavova is partially supported by NSF-DMS, Applied Mathematics program, under Grant # 1516245.

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Malhi, S., Stanislavova, M. When is the energy of the 1D damped Klein-Gordon equation decaying?. Math. Ann. 372, 1459–1479 (2018). https://doi.org/10.1007/s00208-018-1725-5

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  • DOI: https://doi.org/10.1007/s00208-018-1725-5

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